Monte Carlo Methods for Tempo Tracking and Rhythm Quantization

The on tin uous hidden v ariables denote the temp o. Ex-a t omputation of p osterior features su h as the MAP state is in tra table in this mo del lass, so w e in tro du e Mon te Carlo metho ds for in tegration and optimization. The metho ds an b e applied in b oth online and bat h s enarios su h as temp o tra king and trans ription and are th us p oten tially useful in a n um b er of m usi appli ations su h as adaptiv e automati a ompanimen t, s ore t yp esetting and m usi information retriev al. 1. Ho w ev er, when op erating on sampled audio data from p olyphoni a ousti al signals, extra tion of a s ore-lik e des ription is a v ery hallenging auditory s ene analysis task (V er o e, Gardner, & S heirer, 1998). In this pap er, w e fo us on a subproblem in m usi -ir, where w e assume that exa t timing information of notes is a v ailable, for example as a stream of MIDI 1 ev en ts from a digital k eyb oard. One example is automati s ore t yp esetting, 1. Musi al Instrumen ts Digital In terfa e. Ea h time a k ey is pressed, a MIDI k eyb oard generates a short message on taining pit h and k ey v elo it y . In on v en tional m usi notation, the onset time of ea h note is impli itly represen ted b y the um ulativ e sum of durations of previous notes. Durations are en o ded b y simple rational n um b ers (e.g., quarter note, eigh th note), onsequen tly all ev en ts in m usi are pla ed on a dis rete grid. This is due to the fa t that m usi ians in tro du e in ten tional (and unin ten tional) deviations from a me hani al pres ription. F or example timing of ev en ts an b e delib erately dela y ed or pushed. Moreo v er, the temp o an u tuate b y slo wing do wn or a elerating. In fa t, su h deviations are natural asp e ts of expressiv e p erforman e; in the absen e of these, m usi tends to sound rather dull and me hani al. On the other hand, if these deviations are not a oun ted for during trans ription, resulting s ores ha v e often v ery p o or qualit y . Robust and fast quan tization and temp o tra king is also an imp ortan t requiremen t for in tera tiv e p erforman e systems; appli ations that \listen" to a p erformer for generating an a ompanimen t or impro visation in real time (Raphael, 2001b; Thom, 2000). A t last, su h mo dels are also useful in m usi ology for systemati study and hara terization of expressiv e timing b y prin ipled analysis of existing p erforman e data. F rom a theoreti al p ersp e tiv e, sim ultaneous quan tization and temp o tra king is a \ hi k en-and-egg" problem: the quan tization dep ends up on the in tended temp o in terpre-tation and the temp o in terpretation dep ends up on the quan tization. Apparen tly, h uman listeners an resolv e this am biguit y (in most ases) without an y eort.